Let $\mu$ be a Borel probability measure on $R^d$. We say that $\mu$ is an $M_\beta$-measure if its Fourier transformation $\widehat{\mu}$ possesses the following property:
$$\widehat{\mu}(\xi)=o(|\xi|^{-\beta}), |\xi|\to \infty.$$
We define the Fourier dimension of $\mu$ as
$$\dim_F\mu=\sup\{\alpha\in[0,d]:\text{$\mu$ is an $M_{\alpha/2}$-measure }\}.$$
Then it is easy to verify that
$$\dim_F\mu=\liminf\limits_{|\xi|\to \infty}\frac{-2\log |\widehat{\mu}(\xi)|}{\log |\xi|}^ d.$$
Define the Fourier dimension of a Borel set $E\subset R^d$ as
$$\dim_F E=\sup_{\mu\in \mathcal{P}(M)}\dim_F\mu.$$
Fact 1: $\dim_F E\le \dim_H E.$
In fact, recall that , for $s\ge 0$ the $s$-potential at a point $x\in R^d$ is defined as
$$\phi_s(x)=\int \frac{d\mu(y)}{|x-y|^s}.$$
It can be writtten as
$$\phi_s(x)=(|x|^{-s} \ast \mu)(x)=\int |x-y|^{-s}d\mu(y).$$
On the other hand, the Fourier transformation of $|x|^{-s}$ is $c |x|^{s-d}$ (see Some useful facts on Fourier transformation). So, we have
$$\widehat{\phi_s}(\xi)=c|\xi|^{s-d}\cdot \widehat{\mu}(\xi).$$
Then, by Parseval‘s theorem (again, see Some useful facts on Fourier transformation) we can obtain the following expression of $s$-energy:
$$I_s(\mu)=\int\phi_s(x)d\mu(x)=c(2\pi)^d\int |\xi|^{s-d}|\widehat{\mu}(\xi)|^2d\mu(\xi). $$
Now, suppose that $t<\dim_F E.$ Then there exists a measure $\mu\in \mathcal{P}(E)$ such that $|\widehat{\mu}(\xi)|\le b |\xi|^{-t/2}$.
Therefore, if $0<s<t$
$$I_s(\mu)\le c_1 \int_{|\xi|\le 1}|\xi|^{s-d}d\mu +c_2 \int_{|\xi|> 1}|\xi|^{s-d}|\xi|^{-t}d\mu<\infty, $$
which implies that $\dim_H E\ge t.$ So, $\dim_F E\le \dim_H E.$
We say that a Borel set $E$ is Salem set if $\dim_F E= \dim_H E.$
See, Sectorial local non-determinism and the geometry of the Brownian sheet by Khoshnevisan, Wu and Xiao, or Falconer, Chapter 4 in Fractal geometry.
Fact 2: Given $0<s<t<\le 1$, there exists a Borel set $E$ such that $\dim_F E=s$ and $\dim_H E=t.$
See, T. W. Korner, Hausdorff and Fourier dimension, Studia mathematica, 206 (2011), or Christian Bluhm, On a theorem of Kaufman: Cantor-type construction of linear fractal Salem set, Ark. Mat., 36 (1998).
原文：http://www.cnblogs.com/jinjun/p/5042090.html